NCEA Pāngarau Co-requisite (NZQA Standard 32412) common mistakes
NCEA Pāngarau Co-requisite (NZQA Standard 32412) · Understand the traps and how to avoid them.
When converting between units, the candidate multiplies where they should divide (or vice versa), for example converting 4500 mm to metres by writing 4500 × 100 = 450000 m, or converting 2.5 kg to grams as 2.5 ÷ 1000 = 0.0025 g.
They memorise a single rule such as 'times by 1000' without anchoring it to the direction of the conversion. Because both metric and time conversions involve a factor, they treat the factor as directionless and pick whichever operation feels natural, often defaulting to multiplication because it appears more often in practice.
What to do instead
Have them always write the unit fraction with the target on top, e.g. 4500 mm × (1 m / 1000 mm), and cancel the units before computing. Then sanity-check with a known benchmark: a metre is bigger than a millimetre, so the number of metres must be smaller. Practise a mixed set of ten conversions where five go to a smaller unit and five to a larger unit, and require a one-line reason for the direction each time.
The candidate computes a percentage change by taking the percentage of the new value, or adds the percentage to the original without adjusting the base, e.g. a $80 item increased by 15% computed as 80 + (0.15 × 92) or a discount then GST applied in the wrong order and treated as giving the same answer.
Percentage work is often learned as a procedure ('find the percentage, add it on') without a firm grasp of the base quantity. When a problem has two successive changes, the candidate does not track which amount is the current whole, and the multiplicative structure is lost.
What to do instead
Insist on a multiplier method: an increase of 15% means multiply by 1.15, a decrease of 15% means multiply by 0.85, and successive changes multiply in sequence, e.g. 80 × 0.85 × 1.15. Have them state the base in words before each calculation ('15% of the original price'). Include a contrast pair where the order of a 20% rise and 20% fall is swapped, so they see the result depends on the base, not the order.
When solving a linear equation, the candidate moves a term to the other side without changing its sign, e.g. solving 3x + 7 = 22 by writing 3x = 22 + 7, or expanding a bracket as 4(x − 3) = 4x − 3.
The candidate treats equation solving as a set of moves to memorise rather than as maintaining balance. The equals sign is read as 'the answer comes next' rather than as a statement of equality, so the operation applied to one side is not mirrored on the other.
What to do instead
Require the balance model to be written out for the first several problems: subtract 7 from both sides, show the subtraction under both sides, then divide both sides by 3. For brackets, have them draw arrows from the outside term to each inside term before writing anything. Give a checking habit: substitute the answer back into the original equation and verify both sides match, which immediately exposes sign errors.
The candidate rounds too early in a multi-step calculation, for example rounding an intermediate division to two decimal places and then multiplying, so the final answer is outside the acceptable range, or they give an answer with inappropriate precision such as 7.333333 kg for a bag of flour.
School arithmetic rewards writing a tidy answer at every step, and the candidate carries that habit into problems where the intermediate value is not the answer. There is also no working model of how rounding error accumulates across operations.
What to do instead
Teach a two-line discipline: keep full precision in the calculator or in the written fraction until the final step, then round once. Have them annotate the final line with the rounding decision and a reason ('nearest dollar because it is money', 'one decimal place because the measurements are to one decimal place'). Give a comparison task where the same problem is done with early and late rounding so the size of the discrepancy is visible.
When finding the area or perimeter of a composite shape, the candidate uses the slanted side length as the height of a triangle or parallelogram, or adds all the given side lengths when only some are needed, producing a perimeter that includes an internal dividing line.
The candidate has a formula-matching strategy: they scan for a shape name, recall a formula, and substitute whatever numbers are visible. They have not internalised that height means perpendicular distance to the base, and they do not distinguish the boundary of a region from the lines drawn inside it.
What to do instead
Have them redraw the shape and mark the perpendicular height with a right-angle symbol before substituting anything, and identify the boundary with a coloured outline for perimeter questions. For composite shapes, require the decomposition to be sketched and each part labelled with its own dimensions, then a written statement of which lengths are added and which are not. Use a grid-paper task where the answer can be counted, so the formula result can be checked against a visual count.
The candidate reads a graph or table and reports the wrong quantity, for example reading the frequency instead of the value, taking the highest bar as the mean, or describing a trend as 'increasing' when the axis is not starting at zero and the actual change is small.
Graph and table interpretation is treated as a scanning task rather than a structured reading. The candidate does not check axis labels, scales, or units first, and confuses the summary statistics they have memorised with what the display actually shows.
What to do instead
Impose a fixed reading routine before answering: title, both axis labels and units, scale increments, then the specific data point. Require the candidate to write the value with its unit and category in a full sentence before choosing an option. For trend questions, have them quote two actual data values and compute the difference so the claim is supported by numbers rather than by the shape of the line.
The candidate calculates a probability that exceeds 1 or is negative, or adds probabilities of overlapping events without subtracting the overlap, e.g. reporting P(red or blue) as 0.6 + 0.5 = 1.1 for a spinner where the outcomes are not mutually exclusive as described.
The candidate applies an addition rule mechanically without checking whether the events can happen together, and has no reflex that probability is bounded between 0 and 1. The complement idea is underused, so there is no quick internal check available.
What to do instead
Make the sample space explicit: list or tabulate all outcomes before computing, then count favourable outcomes over total outcomes for simple cases. Teach the check that any probability answer must lie between 0 and 1, and that the probabilities of all exhaustive outcomes sum to 1. For combined events, have them state in words whether the two events can occur at the same time before deciding to add or to subtract the overlap.
The candidate produces a correct number but no interpretable answer to the question asked, for example answering a 'how many buses are needed for 250 students at 48 per bus' problem with 5.2 buses, or giving a speed in the wrong unit for the context.
The candidate treats the context as decoration around a calculation and stops when the arithmetic is finished. They have not been asked to re-read the question and check that the answer is a sensible quantity in the real situation.
What to do instead
Require a final sentence in the candidate's own words that answers the question with units and a decision, e.g. '6 buses are needed because 5 buses carry only 240 students'. Build a rounding-to-context drill with cases where rounding up, rounding down, and rounding to nearest are each correct, and have the candidate justify the choice. Insist that the final line of working names the quantity being reported.
When asked to justify or explain reasoning, the candidate restates the calculation ('I multiplied 12 by 8 and got 96') instead of giving the mathematical reason, or writes a conclusion with no supporting evidence, so communication marks are lost even when the answer is right.
The candidate has practised producing answers, not explanations, and does not know what a sufficient justification looks like. They also lack the vocabulary to name the property or relationship being used, so they describe the keystrokes instead.
What to do instead
Provide a sentence frame and practise it: 'I know ... because ... , therefore ...'. Model two examples, one strong and one weak, and have the candidate identify what the weak one is missing. Require that every justification names a mathematical idea (place value, equivalent fractions, the relationship between speed, distance and time) and cites the specific numbers used.
The candidate works in the wrong units within a rate or ratio problem, mixing minutes and hours, or grams and kilograms, so a speed comes out ten or sixty times too large, and they do not notice because the number looks plausible.
Rates combine two quantities and the candidate focuses on the numerical division while ignoring the units attached to each quantity. Because the arithmetic is correct, nothing signals an error, and there is no habit of estimating the expected size of the answer first.
What to do instead
Require units to be written beside every number throughout the working, and converted to a single consistent system before any calculation. Teach an estimation step: 'a car travels about 100 km in an hour, so a 250 km trip should take about 2.5 hours' - then compare the computed answer to the estimate. Use a mismatched-units exercise where the candidate must spot and correct the error in a worked solution.